Should I use my redundant feature as an auxiliary output or as another input feature?
For extra input that does not matter, you should not input it to the network.


  Feature selection, the process of finding and selecting the most
  useful features in a dataset, is a crucial step of the machine
  learning pipeline. Unnecessary features decrease training speed,
  decrease model interpretability, and, most importantly, decrease
  generalization performance on the test set.


Source: A Feature Selection Tool for Machine Learning in Python

As the source says, unnecessary features decreases accuracy and training speed. Moreover, they have no mapping to the labels, so they won't end up being used. They are unnecessary and adding them will only cause you trouble. Hope this helps you and have a nice day!